By A. Dold, B. Eckmann, J.-M. Morel, F. Takens, B. Teissier

ISBN-10: 3540042288

ISBN-13: 9783540042280

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K. for solving equations but not for verifying identities. For example, in solving an equation we frequently move things from one side of the equation to the other. But when we're verifying an identity, we're trying to prove that the two expressions on either side of the equals sign are identical, and we can't possibly do that if we move things from one side to the other. Again in solving an equation we often multiply both sides by some quantity, but in verifying an identity we can't do that, as it changes the values of the expression that we're trying to prove equal.

My teacher was asked a question to which he did not have a good answer, and he wanted us to do things the way ALGEBRA AND TRIGONOMETRY 27 they were done in the book, so found an answer which had satisfied students. Whether he believed it or not, I do not know. I suspect that he never thought about this question, at least not seriously. The author of the current text was probably educated from books like I used, which had the claim about working on one side but no reason why. This was not a satisfactory state of affairs after the introduction of many reasons in the New Math, so something had to be said.

27. Floored by an Olympiad problem The following problem appeared on the first paper in the 1992 Australian Mathematical Olympiad. The solution is due to a student using it as a practice problem. Problem. Let n be a positive integer. Determine how many real numbers x with 1 ≤ x < n satisfy x3 − x3 = (x − x )3 . Solution. Let x = a + b where a ∈ {1, 2, . , n − 1} and 0 ≤ b < 1, Then x3 = a3 + 3a2 b + 3ab2 + b3 so that, since 0 ≤ b3 < 1, we must have one of the three possibilities (i) x 3 = a3 , (ii) x3 = a3 + 3a2 b and (iii) x3 = a3 + 3a2 b + 3ab2 .

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Kuramochi Boundaries of Riemann Surfaces by A. Dold, B. Eckmann, J.-M. Morel, F. Takens, B. Teissier

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